# Smooth functions tangent to the leaves of a foliation

Given two smooth manifolds $M$ and $N$, it is known that if $M$ is compact, then $C^\infty(M,N)$ is a Fréchet manifold whose tangent space at $f \in C^\infty(M,N)$ is the space $$T_f C^\infty(M,N) = \Gamma(M,f^*TN)$$ of all smooth vector fields $\xi(x) \in T_{f(x)}N$ along $f$.

Suppose we are given a foliation $\mathcal{F}$ of codimension $q$ on $N$ and let $C^\infty_{\mathcal{F}}(M,N)$ be the subset of somooth functions $f:M \to N$ tangent to the leaves. Is $C^\infty_{\mathcal{F}}(M,N)$ still a Fréchet manifold? If it is the case, what is its tangent space at a point?

If it is too general to answer, can we at least say something about the case where $\mathcal{F}$ is a foliation of codimension $1$?

• What does it mean for a function to be tangent to the leaves? Is it that the image of the tangent map is always tangent to a leaf? Thanks - Michael – Michael Murray Oct 6 '12 at 14:31
• Isn't that equivalent to the image being contained in a single leaf? – Igor Khavkine Oct 7 '12 at 15:25
• Yes Igor, you're right. – HYL Oct 7 '12 at 20:16

I don't know about the Frechet structure, but can say a few words about the tangent space: if $f_t:M\to N$ is a time dependent family of maps contained in the leaves of $N$ for each $t$, then $\frac{d}{dt} f^*(\alpha)=0$ for any one-form $\alpha \in \Omega^1(N)$ vanishing on the leaves. So a necessary condition for a relative vector field $X$ along a map $f:M\to N$ to be a tangent vector in the space $C^\infty_{\mathcal{L}}(M,N)$ is $L_X(\alpha)=0$ for all one forms $\alpha$ vanishing on the foliation (here $L_X$ is the lie derivative along relative fields which may be defined by $d\circ i_x+i_x\circ d$).
• You are right. I realize that I wasn't distinguishing clearly between the space of maps $S^1\to M$ tangent to leaves, and the moduli space of leaves of the foliation. Thinking it over I would guess that the latter is connected and one dimensional space, while the former has several connected components. Example: two component consisting of maps winding around the middle circle with winding number +1 or -1, which cannot be deformed outside of this leave. But maps winding around other leaves with winding number 1 can be deformed into a map winding twice around the circle. [cont.] – Michael Bächtold Oct 17 '12 at 11:53